Question: Factor the following expression: $6$ $x^2$ $-23$ $x$ $-4$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(6)}{(-4)} &=& -24 \\ {a} + {b} &=& & & {-23} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-24$ and add them together. Remember, since $-24$ is negative, one of the factors must be negative. The factors that add up to ${-23}$ will be your ${a}$ and ${b}$ When ${a}$ is ${1}$ and ${b}$ is ${-24}$ $ \begin{eqnarray} {ab} &=& ({1})({-24}) &=& -24 \\ {a} + {b} &=& {1} + {-24} &=& -23 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {6}x^2 +{1}x {-24}x {-4} $ Group the terms so that there is a common factor in each group: $ ({6}x^2 +{1}x) + ({-24}x {-4}) $ Factor out the common factors: $ x(6x + 1) - 4(6x + 1) $ Notice how $(6x + 1)$ has become a common factor. Factor this out to find the answer. $(6x + 1)(x - 4)$